The content starts by introducing the concept of weakly convex functions, which are a generalization of convex functions. It then discusses proximal subdifferentials, which are a suitable tool for defining criticality for weakly convex functions.
The main contributions of the work are:
Providing sufficient and necessary conditions for the sum rule of the global proximal ε-subdifferentials for the sum of two ρ-weakly convex functions. The modulus of proximal subdifferentiability and the modulus of weak convexity ρ are incorporated into the calculus rules.
Investigating the relationship between the ε-proximal operator of a ρ-weakly convex function f and the ε-proximal subdifferential of f, using the derived calculus rules.
Relating the notion of inexact (approximate) proximal point to Type-1 and Type-2 approximations proposed in the convex setting.
The work also provides several auxiliary results, such as the globalization property of proximal subdifferentials for paraconvex functions, and the characterization of weakly convex functions as the difference between a convex and a quadratic function.
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by Ewa Bednarcz... at arxiv.org 04-24-2024
https://arxiv.org/pdf/2211.14525.pdfDeeper Inquiries